A solid ball of density `rho_(1)` and radius r falls vertically through a liquid of density `rho_(2)`. Assume that the viscous force acting on the ball is `F = krv`, where `k` is a constant and v its velocity. What is the terminal velocity of the ball ?

Options

A. `(4 pi g r^(2) (rho_(1) - rho_(2))/(3k))`
B. `(2 pi r (rho_(1) - rho_(2))/(2gk))`
C. `(2 pi g (rho_(1) + rho_(2))/(3gr^(2)k))`
D. None of these

Answer

To find the terminal velocity of a solid ball falling through a liquid, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Ball**:
- The ball experiences three main forces:
- Weight (downward): \( W = mg \)
- Upthrust (buoyant force, upward): \( F_u = \rho_2 V g \)
- Viscous force (upward): \( F_v = kRv \)

2. **Set Up the Equation for Terminal Velocity**:
- At terminal velocity, the net force acting on the ball is zero. Therefore, the weight of the ball is balanced by the sum of the upthrust and the viscous force:
\[ mg = F_u + F_v \]

3. **Express the Forces in Terms of Density and Volume**:
- The weight of the ball can be expressed as:
\[ mg = \rho_1 V g \]
- The upthrust can be expressed as:
\[ F_u = \rho_2 V g \]
- The viscous force is given as:
\[ F_v = kRv_t \]
where \( v_t \) is the terminal velocity.

4. **Substitute the Forces into the Equation**:
- Substitute the expressions for weight, upthrust, and viscous force into the balance equation:
\[ \rho_1 V g = \rho_2 V g + kRv_t \]

5. **Rearrange the Equation**:
- Rearranging gives:
\[ kRv_t = \rho_1 V g - \rho_2 V g \]
- This simplifies to:
\[ kRv_t = (\rho_1 - \rho_2) V g \]

6. **Solve for Terminal Velocity \( v_t \)**:
- Now, isolate \( v_t \):
\[ v_t = \frac{(\rho_1 - \rho_2) V g}{kR} \]

7. **Substitute the Volume of the Ball**:
- The volume \( V \) of a solid sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
- Substitute this into the equation for \( v_t \):
\[ v_t = \frac{(\rho_1 - \rho_2) \left(\frac{4}{3} \pi r^3\right) g}{kR} \]

8. **Simplify the Expression**:
- The \( R \) cancels out:
\[ v_t = \frac{4}{3} \frac{\pi g r^2 (\rho_1 - \rho_2)}{k} \]

### Final Result:
Thus, the terminal velocity \( v_t \) of the ball is given by:
\[ v_t = \frac{4 \pi g r^2 (\rho_1 - \rho_2)}{3k} \]